Tag Archive | Logic

Proving a Negative

In this post, I’m going to attempt to prove a negative:

1) P → Q

2) ¬Q

3) ∴ ¬P

Wow…that was easy. I wonder if I can take it one step further, and prove a universal negative? Hang on to your hats, cause this one is tricky:

1) ∀xPx → ∀xQx

2) ¬∀xQx

3) ∴ ¬∀xPx

Quod Erat Demonstratum.

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Logic And Reality

Gil Sanders of Walking Christian has responded to my article on logic, math, and presuppositionalism. While his objections are far more well thought out than the non-responses of presuppositionalists, there are a few points I need to clarify about logic.

First, he takes classical logic as something metaphysical. This is fine in a general sense, but as I have mentioned in my previous article, classical logic doesn’t always match up perfectly with reality (here I am taking metaphysics as a discipline to be something which attempts to describe the features of reality). I cited relevant logic as an example of this – material implication just does not capture reality all the time* (hardly an obscure problem!).

Gil writes, “A language could even create a contradictory syntax and strangely arrive at some ”coherent” function but at that point, it just gets ridiculous. If that’s all “conditional logics, relevant logics, paraconsistent logics, free logics, quantum logics, fuzzy logics” do then I think they’re useless in telling us what reality is.”

He is actually touching on an important point here – not all logics are useful. Elsewhere on my blog I’ve mentioned Douglas Hofstadter’s MU puzzle, and said that it’s possible to define a logic with only one theorem. While we can do this, there’s really no utility in doing so. But there are some nonclassical logics that are highly useful, and do seem to say something about reality in a sense. The relevant logic example is only one – there is also ternary logic. The database language SQL implements ternary logic, and we’ve even built ternary computers a few times in the past.

A third example is fuzzy logic. I’ll simply say this – I challenge Gil to solve the Sorites paradox using only classical logic.

Now, as for classical logic, I do think it is useful at describing reality – most of the time. I see classical logic as analogous to regular, “everyday” English, with nonclassical logics being analogous to the various technical jargons of different fields. This doesn’t seem exactly right to me, but it’s the best analogy I can think of. One might say that technical jargons are merely extensions of everyday English, but I don’t think this is the case, because everyday English includes many slang terms not acceptable in technical fields; thus there is some exclusivity.

Now, this article is admittedly a bit short, and doesn’t touch on every point. This is intentional. I could write pages and pages and pages about logic, with hundreds of citations, but that really won’t get us anywhere. Rather, this is just to continue a back and forth discussion about logic with Gil, and I would suggest that we focus specifically on the paradox of material implication and the Sorites paradox, and the failure of classical logic to explain these things well.

 

*addendum: this can be seen in the truth table for p → q (http://en.wikipedia.org/wiki/Material_conditional#Truth_table). When p = True and q = True, p → q = true. This leads to natural language problems where “If the sun is a star, then cheese is made from milk” is true. But this obviously doesn’t follow. For this to make sense, there needs to be a causal connection that classical logic just can’t capture.

The Presup Challenge

The Presup Challenge:

Define a formal logic, complete with syntax, semantics, a deductive system, and a meta-theory. Then we can talk about whether it’s absolute or not.

UPDATE: The Beginner Presup Challenge

Complete this logic practice quiz:

Which of the following are well-formed formulas:

1. ¬¬P

2. (¬P ↔Q ⊃ R)

3. (Q v (P ↔R))

4. (P ∧ R)

5. (P ∨ Q)

Construct truth tables for the following:

1. ¬

2. ↔

3. ⊃

4. C

5. ∨

6. ⊻

Conduct a Moorean Shift on the following:

(P ∨ Q) ⊃R

R

∴ (P ∨ Q)

Conversation With Sye Ten Bruggencate

This is a verbatim personal correspondence I had with Sye Ten Bruggencate via email dated 28 April 2012, regarding my blog post found here. Honestly, after this I’m so frustrated by the inanity of it all that I’m not going to debate logic with someone unless they’ve at least read an introduction to logic text. I don’t have the time or the patience for stuff like this. 

Me:

Sye,

I’ve recently written a critique of the presuppositionalist analysis of logic on my blog that you may be interested in. You’re welcome to respond here: https://dubitodeus.wordpress.com/2012/04/23/logic-math-and-presuppositionalism/

Enjoy! 🙂

Sye: Sorry, but I had to stop at “logic is conventional.” 

Me: Umm…what? Are you telling me that you’re not even going to finish reading my response to the core of your apologetic strategy?

Sye: Nah, when you said that logic was conventional, I lost all interest.  You see, I could just make a convention that everything you wrote is illogical, and be done with it.

Cheers.

Me: See, if you had read my article, you would understand why what you said is completely irrelevant to the topic. But since you don’t seem to have an interest in actually discussing what I wrote, let’s play this game your way:

How do you account for elliptic geometry?

Sye: With Friday motballs under the twice.  (I just made a new convention of logic  🙂

Me: Ok Sye, I have to ask…are you interested in a conversation, or just in being obtuse and contrary?

Sye: I am simply answering a fool according to his folly (Proverbs 26:5).  If logic is conventional, you can have no problem with my response,but you do, exposing the fallacy of your view.

Me: If logic is conventional, you can have no problem with my response”. See, once again, you completely misunderstand conventionalism with regard to logic. What you just said is like responding to a question said in French with, “That doesn’t mean anything. If language is conventional, you can have no problem with my response.”

Just because logics are a convention, doesn’t mean that you can arbitrarily say silly things and then declare that everyone must accept them as not silly. You can create a new convention, but you still have to define rules for that convention. Just like languages – you can create a new language, but you still have to define a grammar. Just like board games – you can create a new board game, but you still have to write down the rules for your game.

But once again, if you read my article, you would already know this. I’d recommend that you read a logic textbook, but if you can’t even read a few hundred words on the subject without deciding to be obtuse, I guess there’s not much hope for that. It’s obvious that you don’t understand logic, but I’m starting to suspect that you don’t want to understand it.

Sye: //”Just like languages – you can create a new language, but you still have to define a grammar.”//

Not according to my new convention.

Me: This is pointless. My blog post renders everything you’ve said irrelevant, and brings up huge problems for your account of logic, yet you refuse to read it. So, do you mind if I post this conversation publicly, so everyone can decide for themselves who “won”?

Sye: Please do.

Rick Warden’s Argument For Theism

Blogger Rick Warden has decried the fact that the “top 20 atheist bloggers” have declined to offer a response to his argument for theism[1]. I think that most arguments, even if they are poor, deserve a response; so I will attempt to answer his argument[2] here (the picture is of course just a joke). The first section reads as follows:

I. Formal logic presupposes certain truths theoretically exist as a basis for sound reasoning.
A. A categorical syllogism, for example, requires the existence of implied universal truth and validity.
1. At least two laws of logic apply in all possible worlds.
a. Law of non-contradiction: It is not possible that something be both true and not true.
b. Law of identity: A = A. Something is what it is and has at least one identifying characteristic.

His (I) is correct – any logical system will take for granted that certain formulas are true; these are called the axioms of the system. But his support for this is mistaken. A syllogism does not need the existence of universal truth, because the syllogism is only valid within one system. If you take a syllogism formulated in classical logic which uses the law of non-contradiction as a premise, and translate it into a paraconsistent logic, that syllogism will no longer be valid.

He’s also mistaken about his definitions of the “laws” of logic”. They are not written in English, but in the language of symbolic logic. The LNC reads, “~(A & ~A)”, and the law of identity reads, “A → A”. A proper translation of these into English would read, “not (A and not-A)”, and “A materially implies A”.

Furthermore, the only reason we don’t allow a contradiction in “classical” logic is because that, given the rules of that system, a contradiction makes every formula trivially true – thus rendering the system useless. But if we create a new system by removing the rule of inference called “disjunctive syllogism”, then this doesn’t happen, and we can have contradictions without rendering it useless.

But anyway, I’ll grant (I), with the caveat that which “truths” (axioms) are presupposed is going to depend on what system you’re working in.

II. The foundation of cohesive logic appears to have been undermined by quantum physics.
A. A quantum particle has ambiguous identifying characteristics until it is measured and collapsed.
B. Quantum non-locality and entanglement imply boundaries that were assumed to be finite and localized are not.
C. QM phenomena and influences are not neatly compartmentalized apart from the Visible day-to-day World
D. If the physical world is truly interconnected by energy, there is only one implied physical identity.
E. It is not the laws of physics that determine how information behaves in our Universe, but the other way around.

Two things to say here. First, I disagree. Maybe it’s true that “classical” logic can’t describe the way things behave on the quantum level, but so what? Such situations are one of the reasons why we have other logics to work with.

Second, Rick mentions a lot of stuff in his writing regarding this point that doesn’t even appear to be relevant. In addition to talking about QM, he criticizes materialism and Ayn Rand’s objectivism. But some atheists are not materialists, and most are not objectivists. I am neither.

III. NDE Cases Support a Cohesive, Logical Understanding within a Theistic Framework.
A. NDE patients describe situations they could not have perceived with their physical senses.
B. Reynolds described the appearance of a unique instrument used and recalled a specific conversation.
C. A Dutch NDE patient described aspects of an operation that occurred observed during clinical death with a cardiac arrest.
D. People born blind have made accurate, detailed descriptions of images they could not have seen with their natural eyes.
E. A specific identity and locality is maintained while experiencing clinical death, consistent with the law of identity.
F. NDE accounts imply that human volition (free will) exists and operates on a spiritual level.
G. NDE accounts imply a God with a loving nature exists. This supports the theist view over other religions.

NDEs certainly do point to some strange things which are difficult to explain, but they don’t necessarily point away from atheism. It may be that these experiences are completely naturalistic, and merely point to the fact that perhaps our senses do work when we currently think they don’t, and these experiences are merely the illusion of having a shift in location. Or it may be that substance dualism is indeed true.

But in any case, the best this can do is shift the probabilities away from materialism. These probabilities would be then redistributed equally, raising the probability of all other possibilities – theism, non-materialistic atheism, solipsism, etc. So this isn’t a gain in likelihood for theism compared to atheism; just compared to materialism.

Also, I don’t see why (F) is true. How do NDEs say anything about free will at all?

IV. Materialism has failed to provide support for answers to foundational questions while theism has provided such support.
A. Universal and certain truth and validity are implied as a necessary combination in making formal philosophical arguments but the possibility of absolute truth is rejected by most materialists because of the theistic implications.
B. Studies in quantum physics offer metaphysical under-determinism while cohesive logic regarding identity remains beyond reach.
F. Materialism has Failed to provide minimal answers with regard to the origin of the universe, the origin of matter, the origin of life, the origin of information, the origin and makeup of consciousness.
G. Theism does provide a logical and cohesive framework and specific answers to the above questions in keeping with related evidence.

I guess the lettering is off here. Oh well, no matter. Anyway, I feel that I’ve already answered the point about materialism above, so I won’t reiterate it here.

V. Conclusion
A. Proof is affirmed by logic and material evidence and the preponderance of evidence supports a theistic interpretation.
1. The materialist view is logically inconsistent and in conflict with science and evidence implying the supernatural.
2. The Christian view is supported by cohesive logic, science, evidence and scriptural text.
a. Hebrew 11.3: Logic, information and the spiritual dimension form the basis of prime reality.
b. John 1.1, 1.14: God is the logical basis of prime reality.
c. Colossians 1.17: God is both the creator and enabler of the physical world.

Rick makes an odd move here from theism to Christianity. I can’t find where Christianity suddenly jumps in, given that he’s only been talking about theism this whole time. He also once again critiques materialism, which is not identical to atheism.

Anyway, there’s a few things to say about all this. First, it seems like I could grant all his premises, and still consistently be an atheist. None of the premises given clash with atheism – just with materialism, objectivism, etc.

Second, this seems to be not so much an argument, as a series of somewhat related statements. No rules of inference are given, and I struggle to think of any that could produce his conclusions with the premises he has.

Finally, I’d like to distance myself from at least some of the atheists who have refused to respond to Rick’s argument. I’m not familiar with all of them, but P.Z. Myers is just another typical “new atheist”; and John Loftus is quite unreasonable (just ask Victor Reppert!)

But, I’d be more than happy to re-examine this argument if he wants to reformulate it, or provide his inferences. I also invite him to respond to the arguments for atheism I’ve provided elsewhere on this blog.

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[1] http://templestream.blogspot.com/2011/07/top-20-atheist-bloggers-decline.html

[2] http://templestream.blogspot.com/2011/03/how-identity-logic-and-physics-prove.html

Logic, Math, and Presuppositionalism

Theists who employ what is known as “presuppositional apologetics”, and more specifically, the transcendental argument for God, often make confused claims about logic. John Frame claims that logic is based on the nature of God[1], and Matt Slick says that the laws of logic are absolute and independent of the human mind[2]. There are others who use such argumentation in slightly different ways (such as Eric Hovind and Sye Tenbruggenate), but the general view seems to be that theism, and specifically Christianity, is needed in order to justify one’s use of logic.

But this view of logic is sorely mistaken. The so-called “laws of logic” are merely conventions that exist within a man-made, formal system. To see why, we merely have to look at “logic” for what it really is – an entire field dedicated to the study of inference and relation. In the past century, many methods of doing this have appeared. Graham Priest writes:

Despite this, many of the most interesting developments in logic in the last forty years, especially in philosophy, have occurred in quite different areas: intuitionism, conditional logics, relevant logics, paraconsistent logics, free logics, quantum logics, fuzzy logics, and so on. These are all logics which are intended either to supplement classical logic, or else to replace it where it goes wrong.[3]

Relevant logic, for example, differs from classical logic in that it attempts to solve the paradox of material implication. The paradox of material implication is the observation that “If the moon is made of cheese, then 2+2=4” is true, even though the antecedent is in no way related to the consequent. Material implication just does not capture what we really mean by “if…then” – yet it is classically valid. Relevant logic attempts to solve this by saying that andecedents must be “relevant” to consequents.

This seems to make sense, but it leads to an interesting feature of relevant logic. Relevant logic is not explosive, which means that unlike classical logic, you can sometimes have contradictions which do not entail the trivial truth of every proposition. Thus, “~(A & ~A)”(the so-called “law of non-contradiction”) is not a theorem of relevant logic (while it is a theorem of classical logic).

This is important because it shows that the unchanging, transcendent view of logic that presuppositionalists talk about just isn’t the case. These logical and mathematic systems are just models we invent to examine different ideas. We can and have changed them, or even thrown them out and started over with different “laws”, many times in the past. Relevant logic is just one example. Some other logics even change the definition of truth. Four-valued logics (often used in computing and electronics) don’t have “P v ~P” as a theorem; and fuzzy logics even have an infinite range f truth values. Intuitionist logics aren’t even concerned with truth, but justification. And these all have real-world applications.

Even mathematics does this. In elliptic geometry, for example, the sum of the angles of a triangle is more than 180 degrees; and Euclid’s parallel postulate is false – there are no parallel lines in elliptic geometry. Now, one might object by saying, “well, that’s just a thought experiment, and doesn’t obtain in reality”. The problem with this is that neither does Euclidean  geometry. When’s the last time you saw a triangle? You haven’t. You never have. The only thing that exists in reality is an approximation of a triangle. A “real” triangle would have to have infinitely thin sides in order to have angles that add up to exactly 180 degrees; and would also have to be infinitely flat. We live in a 3-dimensional (at least) universe, and triangles exist in 2 dimensions. A triangle is just a concept, a thought experiment.

All this points to a simple fact: logic and mathematics are made up. We invented them to describe what we see, and they are only approximations.
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[1] http://www.reformed.org/master/index.html?mainframe=/apologetics/martin/frame_contra_martin.html

[2] http://carm.org/transcendental-argument

[3] Priest, Graham. An Introduction to Non-Classical Logic. 2nd ed. New York: Cambridge University Press, 2008. xvii.